The Research On The Equivalence Of N-D Polynomial Matrices

The equivalence of multivariate polynomial matrices is closely related to the equivalence of multidimensional systems.It is widely used in mathematics and engineering fields.The equivalence of polynomial matrix and its Smith normal form is one of the most frequently considered types in the multivariate polynomial equivalence problem.Smith normal form is simple,which can simplify the analysis and numerical calculation of the system.The main contents of this paper are as follows:Firstly,we investigate a class of matrices which be of normal rank r and whose greatest common divisor of maximal order minors is?z₁-f?z ₁,z ₂,?43?,z _n??^qon polyno-mial rings.By using the method of"hierarchical recursive",we shall give some necessary and sufficient conditions that the ideal generated by the reduced k-th order minors is unit ideal,under which matrices are equivalent to its Smith normal form.These conditions can be easily judged by calculating these ideal Gr?bner bases.Secondly,based on the definitions of equivalence and similarity of matrix,we shall give the necessary and sufficient conditions for the equivalence of two matrices on the ring of binary polynomials,and the equivalence between a class of special multivariate polynomial matrices and their Smith normal form is discussed by using this condition.